Sp(2)/U(1) and a Positive Curvature Problem

TL;DR

This paper demonstrates that the coset space Sp(2)/U(1) admits metrics with positive curvature for commuting pairs, providing a unique example that distinguishes between general positive curvature and positive curvature only for commuting pairs.

Contribution

It proves the existence of positively curved metrics for commuting pairs on Sp(2)/U(1), highlighting a new class of spaces with this specific curvature property.

Findings

Sp(2)/U(1) admits positively curved metrics for commuting pairs

This is the first example of a space with positive curvature only for commuting pairs

Contrasts with Wilking's result on the non-existence of positive curvature in the general sense

Abstract

A compact Riemannian homogeneous space $G/H$, with a bi--invariant orthogonal decomposition $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ is called positively curved for commuting pairs, if the sectional curvature vanishes for any tangent plane in $T_{eH}(G/H)$ spanned by a linearly independent commuting pair in $\mathfrak{m}$. In this paper,we will prove that on the coset space $\mathrm{Sp}(2)/\mathrm{U}(1)$, in which $\mathrm{U}(1)$ corresponds to a short root, admits positively curved metrics for commuting pairs. B. Wilking recently proved that this $\mathrm{Sp}(2)/\mathrm{U}(1)$ can not be positively curved in the general sense. This is the first example to distinguish the set of compact coset spaces admitting positively curved metrics, and that for metrics positively curved only for commuting pairs.